Abstract

Steering, a quantum property stronger than entanglement but weaker than non-locality in the quantum correlation hierarchy, is a key resource for one-sided device-independent quantum key distribution applications, in which only one of the communicating parties is trusted. A fine-grained steering inequality was introduced in [PRA 90 050305(R) (2014)], enabling for the first time the detection of steering in all steerable two-qubit Werner states using only two measurement settings. Here we numerically and experimentally investigate this inequality for generalized Werner states and successfully detect steerability in a wide range of two-photon polarization-entangled Bell local states generated by a parametric down-conversion source.

Introduction

The notion of non-locality was first introduced in 1935 by Einstein, Podolsky and Rosen Einstein et al. (1935) by discussing a “spooky action at a distance”. This led Schrödinger to introduce the concept of steering, in his response Schrödinger (1935, 1936) to the EPR paper. The term steering describes a property of quantum mechanics that allows the subsystems of a bipartite entangled state to affect each other’s state upon measurement: indeed the measurement of one of the subsystems can project (or steer) the other subsystem in a particular state, which depends on the chosen measurement on the first subsystem. In 2007, this concept was reformulated in terms of a quantum information task by Wiseman, Jones and Doherty Wiseman et al. (2007), which allowed them to establish a strict hierarchy between quantum correlations of increasing strength: entanglement, steering and Bell non-locality. Seen in this light, steering is the resource that allows Alice to win a game in which she tries to convince Bob (who does not trust her) that she can prepare an entangled state and share it between them.

As was shown in 2012 by Branciard et al.Branciard et al. (2012), this resource can be exploited for quantum cryptography in a one-sided device-independent quantum-key-distribution (1SDI-QKD) scenario, intermediate between standard QKD in which both parties need to trust their measurement apparatus and device-independent QKD (DI-QKD) Ekert (1991); Acín et al. (2007) in which neither does: for 1SDI-QKD, only one of the parties needs a trusted measurement device. Compared to the DI-QKD scenario, this additional constraint of one trusted device must be fulfilled but, in return, the security can be based on the violation of a steering inequality rather than a Bell inequality, which lowers significantly the experimental requirements in terms of detection and transmission efficiencies Wittmann et al. (2012); Bennet et al. (2012); Smith et al. (2012) because, in particular, the detection loophole needs to be closed only on Alice’s side and the noise tolerance is higher.

In this article, we investigate numerically and experimentally a recently proposed steering inequality Pramanik et al. (2014) based on fine-grained uncertainty relations Oppenheim and Wehner (2010), which is more efficient in that it allows the detection of steerability in a large range of noisy two-qubit states, in particular in all steerable Werner states, with two measurement settings. We first recall the fine-grained inequality in Section I. In Section II we apply the inequality to generalized Werner states, give a procedure to correctly choose the measurement settings and compare its performance with coarse-grained inequalities. Finally in Section III we illustrate the procedure with a simple experiment.

I Fine-grained steering inequality

Let us first recall the fine-grained steering inequality introduced in Ref. Pramanik et al. (2014) and the underlying game scenario. Alice prepares a bipartite state ρAB, keeps the part labeled A for herself and sends the part labeled B to Bob. Bob then asks Alice to steer B to any eigenstate of an observable UB, randomly chosen in the set {P,Q} (with P and Q maximally incompatible). Alice measures A with an observable UA (with UA=S if UB=P and UA=T if UB=Q) and sends her measurement outcome a∈{0;1} to Bob. Bob finally measures B with the previously chosen observable UB and gets an outcome b∈{0;1}. After repeating these steps a large number of times, Bob is convinced that Alice can indeed steer his subsystem (i.e. that ρAB is steerable) if the following inequality is violated:

Fa,b

=12(P(bP|aS)+P(bQ|aT))

(1)

≤Flim=12maxP∗,Q∗[maxλ[Pq(bP∗|λ)+Pq(bQ∗|λ)]]

where P(bP,Q|aS,T) is the conditional probability of Bob getting the outcome b upon measurement of P or Q when Alice announces the outcome a, P∗ and Q∗ range over all possible maximally incompatible measurements, and Pq(b|λ) is the probability of obtaining an outcome b upon a quantum measurement on a quantum system ρB described by a hidden variable λ. The left-hand side Fa,b of the inequality is the steering parameter which can be estimated from measurement results. The right-hand side Flim is calculated by considering a local hidden state (LHS) model for ρB: this model corresponds to the cheating strategy of a dishonest Alice who prepares a local state ρB instead of an entangled state ρAB and announces values of a that do not correspond to actual quantum measurements. As was shown in Ref. Pramanik et al. (2014), if Alice knows beforehand the set {P,Q} of Bob’s possible measurement settings (“scenario I”), then Flim(I)=(1+1/√2)/2≈0.854. If, however, Alice does not know the set {P,Q} before preparing ρAB (“scenario II”), then Flim(II)=3/4.

Figure 1: (Color online) Experimental set-up (for more details see QuT ). The main elements of the source are shown in the dashed-line box: a CW laser beam at 405 nm pumps two BBO crystals with orthogonal axes. The angle χ of the half-wave plate (HWP) controls the amount of photon pairs generated in the first and second crystals. At Alice’s and Bob’s station, projective measurements are implemented with a rotating polarizer. The photons at 810 nm are coupled into single-mode fibers and detected in coincidence by silicon avalanche photodiodes (Si-SPAD) and timing electronics. Ceiling lamps (not shown) are used to fix the amount of unpolarized background noise.

Ii Numerical analysis for generalized Werner states

In the following, we study the aforementioned inequality in both scenarios for states of the form:

ρ=p(η∣∣Φ+α⟩⟨Φ+α∣∣+(1−η)∣∣Φ−α⟩⟨Φ−α∣∣)+(1−p)41A⊗1B,

(2)

where ∣∣Φ±α⟩=√α|00⟩AB±eiφ√1−α|11⟩AB is a pure state, α,p,η∈[0;1], 1 is the 2×2 identity matrix and subscripts A,B refer to Alice’s and Bob’s qubit respectively. These states model well polarization-entangled two-photon pairs produced e.g. by bulk type-I spontaneous parametric down-conversion sources Kwiat et al. (1999) (see the dashed-line box in Fig. 1) where a laser beam, with a linear polarization adjusted by a half-wave plate (HWP) set at an angle χ, pumps two adjacent nonlinear crystals having optical axes orthogonal to each other. Then α=cos2(2χ) corresponds to the amount of pairs emitted in the first crystal (horizontally-polarized photons, denoted by |00⟩) with respect to those emitted in the second crystal (vertically-polarized photons, denoted by |11⟩); φ is the phase between the two possible pair emission processes; p is linked to the amount of unpolarized noise present in the set-up (background light, fluorescence of the optical elements…); and η is linked to a dephasing noise accounting for the partial distinguishability between the optical modes of photons emitted by each crystal.

These expressions, for given values of η and α, give lower bounds on the value of p for which ρ is entangled or Bell non-local (see black and red lines in Fig. 2). In particular, for Werner states Werner (1989) (i.e. η=1 and α=1/2), the state is entangled if and only if p>1/3 and it is Bell non-local if p>1/√2.
Note that here we use the CHSH inequality to distinguish between Bell non-local states and states admitting a LHV model (which we call Bell-local here) since we restrict Alice and Bob to only two projective measurement settings Acín et al. (2006). This would not be valid for an arbitrary number of projective measurements since, in that case, it has been shown Acín et al. (2006) that, for Werner states, the separation occurs for p=1/KG(3), KG(3) being Grothendiek’s constant of order 3 for which the best known lower Brierley et al. (2016) and upper Hirsch et al. (2017) bounds up to now are 1.4261≤KG(3)≤1.4644, i.e. 0.7012≥p≥0.6829. Hence, strictly speaking, we are sure that a Werner state is Bell-local only if p<0.6829.

To compute the steering parameter defined in Eq. (1), any combination {a,b} can be chosen with a,b∈{0;1} but some choices may give larger values than others depending on the asymmetry of the state, thus we will define a more general steering parameter F=maxa,b(Fa,b). Writing Bob’s and Alice’s measurement settings as U=cosθUσz+sinθU(cosϕUσx+sinϕUσy) (with U=P,Q,S,T and σx, σy and σz the Pauli matrices), and imposing θQ=θP+π/2 and ϕQ=ϕP such that P and Q are maximally incompatible measurements, F can be expressed as:

F

+

[1+p(−ξSP+(ξ−SP)CT)+ζTCPST)]/[1+pξCT]]

where ξ=σ|2α−1|, ζA=2(2η−1)√α(1−α)cos(φ−ϕA) (A=S,T), CU=cosθU and SU=sinθU (U=P,S,T), and σ denotes the sign of cos(θP+π/4) (see Appendix A for details).
The angles θS, ϕS, θT and ϕT are optimized so as to maximize F once P and Q have been fixed.
For scenario I, Bob’s measurements P and Q are fixed to σz and σx respectively, thus FI=maxS,T(F) with θP=ϕP=ϕQ=0 and θQ=π/2.
For scenario II, P and Q must be chosen such that F is minimized: FII=minP,Q(maxS,T(F)). Note that for Werner states (i.e. α=1/2 and η=1) which are symmetric states, any choice of maximally incompatible settings P and Q will give the same value of F; however for α≠1/2 this is not the case and the minimization must be done in order to avoid overestimating F (see Appendix B). This constraint comes from the requirements that P and Q should be unknown by Alice when she prepares ρAB.

Figure 3: Bounds on p as a function of χ (in degrees), for a generalized Werner state with η=1. The thick red, dark blue, dark green and black lines are the lower bounds presented in Fig. 2. The light green dashed line is an upper bound for non-steerable states found in Ref. Hirsch et al. (2016) by a numerical iterative construction of a LHS model with 4 steps. The light blue dotted line is a lower bound for steerable states found also in Ref. Hirsch et al. (2016) with the SDP method of Ref. Skrzypczyk et al. (2014) and 9 measurement settings. The steering lower bound given by the fine-grained steering inequality in scenario II (dark green line) lies in between these two numerical bounds.

Equation (LABEL:Eq:RhoSteeringPr) gives a lower bound on p for the state ρ to be steerable, according to scenario I or II (see blue and green lines in Fig. 2). For the particular case of Werner states, using scenario I, we find a lower bound of 1/√2 for p, the same as for Bell non-locality detected by the CHSH inequality and the same as for all coarse-grained steering inequalities with two measurement settings Cavalcanti and Skrzypczyk (2017). However, using scenario II, the theoretical bound p=1/2Wiseman et al. (2007) is reached with only two measurement settings.
Note that we can even conjecture that scenario II gives an optimal lower bound for generalized Werner states (i.e. for any α∈]0;1[). Indeed Fig. 3 shows that this bound (dark green full line) lies between a lower bound for steerable states (light blue dotted line) and an upper bound for states with a LHS model (light green dashed line) that were both calculated numerically in Ref. Hirsch et al. (2016), respectively with a semi-definite program Skrzypczyk et al. (2014) and with a new iterative method for constructing LHS models Hirsch et al. (2016); Cavalcanti et al. (2016).
We can also notice that, for both scenarios, the set of steerable states detected with two settings is strictly larger than the set of Bell non-local states seen by CHSH: for α∈]0;12[∪]12;1[, the lower bound on p for steering is strictly lower than the one for Bell non-locality. In particular, even for scenario I, Bell local steerable states can be detected for a large range of values of α.

Figure 4: (Color online) (a) Measured visibilities in the σz (blue dots) and σx (red open circles) bases as a function of the angle χ of the pump’s half-wave plate. Error bars (not shown) are about the size of the symbols or smaller. Solid lines: simulations for p=0.90 and η=1. Dotted lines: simulations for p=0.90 and η=0.96. (b) Bell and (d) steering parameters as a function of α. Symbols: measurements, full lines: simulations for p=0.90 and η=1, dotted lines: simulations for p=0.90 and η=0.96. Bell parameter S: red squares and lines, steering parameters FI: blue circles and lines, and FII: green diamonds and lines. The mixed horizontal lines show the LHV limit Slim=2 and the LHS limits Flim(I)=(1+1/√2)/2 and Flim(II)=3/4. Vertical error bars (not shown) stemming from Poisson photon counting statistics are <0.025 for S (about the symbol size) and <0.005 (α∈[0.1;0.9]) or <0.02 (α<0.1 or α>0.9) for F (see Appendix C). (c) optimized CHSH measurement settings (see Appendix A) B,B′=cos(θB,B′)σz+sin(θB,B′)σx for Bob when fixing Alice’s measurements to A=σz (θA=0) and A′=σx (θ′A=π/2). (e) optimized steering measurement settings for Alice for scenario I (θS in blue and θT in light blue) and for scenario II (θS in green and θT in light green), calculated with Wolfram AlphaWol for p=0.90 and η=0.96 (see Appendix A). In all plots, the vertical mixed line corresponds to α=1/2.

Iii Experimental implementation

We have experimentally tested this fine-grained steering inequality in both scenarios with a commercial source of polarization-entangled two-photon states (QuTools QuT ) based on the scheme of Ref. Kwiat et al. (1999) (see Fig. 1). The projective measurements corresponding to settings UA on Alice’s photon and UB on Bob’s photon, with UA,B=cos(θA,B)σz+sin(θA,B)σx (i.e. in the (σz;σx) plane of Bloch’s sphere), are implemented with a polarizer whose axis is set at an angle θA,B/2 (projection on the +1 eigenstate of UA,B) or θA,B/2+π/2 (projection on the −1 eigenstate of UA,B) with respect to the vertical direction. Note that σy measurements are not needed when φ=0 or π (see Appendix A).

We first characterized the experimental state with visibility measurements Vz (Vx) in the σz (σx) basis, for different values of χ (Fig. 4(a)). Modeling the state with Eq. (2) and using the relation α=cos2(2χ), one can show that Vz=p and Vx=2p(2η−1)√α(1−α) (see Appendix A). We thus deduced a noise parameter p=0.90, a dephasing parameter η=0.96 and a phase φ=π.

In Fig. 4(b) we show the measured value of the Bell parameter S=E(A,B)−E(A′,B)+E(A,B′)+E(A′,B′), with E(A,B)=P(0A,0B)+P(1A,1B)−P(1A,0B)−P(0A,1B), and its theoretical value given by Eq. (4), for the optimized measurement settings A,A′,B,B′Gisin (1991) shown in Fig. 4(c). The state violates the CHSH inequality S≤2 and is thus non-local for α∈]0.075;0.925[, with a maximal value of S=2.45 for α=1/2.
The corresponding measured values of the steering parameter in both scenarios FI and FII are shown in Fig. 4(d). For scenario II, Bob’s measurement angles have been set to ϕP=ϕQ=0 and θP=π/4, θQ=3π/4 which minimize Eq. (LABEL:Eq:RhoSteeringPr) for F (for any value of α, p and η, see Appendix B). For both scenarios, Alice’s settings have been optimized (with ϕS=ϕT=φ=π) for each value of α so as to maximize F and are shown in Fig. 4(e). The state violates the fine-grained steering inequality (Eq. (1)) in scenario I for α∈]0.022;0.978[ and in scenario II for α∈]0.015;0.985[. For scenario I, the maximum value of FI=0.935 is obtained for α=0.35 and α=0.65. For scenario II, the maximum is FII=0.932 for α=1/2.

In Fig. 5, we have plotted the measured values of FI and FII against S, together with the simulations for p=0.90 and η=0.96. We can identify five main zones in this plot, corresponding to quantum correlations of different strength. The shaded red area corresponds to Bell non-local states that violate the CHSH inequality (i.e. S>2) and also violate the fine-grained steering inequality. The blue (green) area corresponds to steerable Bell local states detected in scenario I or II: FI>(1+1/√2)/2 or FII>3/4, and S≤2. The white area corresponds to states that violate neither the CHSH inequality nor the fine-grained steering inequality. Finally the grey area corresponds to states that are Bell non-local but unsteerable which is in contradiction with the established hierarchy of quantum correlations Wiseman et al. (2007); none of our measurement results fall in this zone.

Iv Conclusion

In conclusion, we have numerically and experimentally investigated the performance of the fine-grained steering inequality of Ref. Pramanik et al. (2014) for the detection of steerability in experimentally-relevant two-photon states which can be modeled as generalized Werner states with dephasing. We have shown that contrary to other steering inequalities which are coarse-grained Cavalcanti and Skrzypczyk (2017) and require strictly more than two measurement settings to detect steerability in Bell local states Saunders et al. (2010), this fine-grained inequality is able to detect a much larger set of steerable states with only two measurement settings, in particular all steerable Werner states and (most probably) all steerable generalized Werner states. We have also shown that even using the most conservative LHS bound of scenario I, the inequality allows the detection of steerability in states that do not violate the Bell-CHSH inequality. Finally, for scenario I, a key rate r≥log2[FI/(2Flim(I)−FI)] for 1SDI-QKD has been proven in Ref. Pramanik et al. (2014) against individual eavesdropping attacks; with our best value of FI=0.935, a rate r≥0.276 secure bit per photon detected by Bob could thus be achieved. It will be interesting to extend this calculation to scenario II, which allows for a better noise tolerance.

Acknowledgements

We thank Ioannis Touloupas for his help with the data acquisition, and Nicolas Brunner and Flavien Hirsch for kindly providing numerical data for steering and LHS bounds in Ref. Hirsch et al. (2016). We acknowledge financial support from the BPI France project RISQ, the French National Research Agency projects QRYPTOS and COMB, and the Ile-de-France
Region project QUIN. The work of MK is supported in part by EPSRC grant number EP1N003829/1 Verification of Quantum Technology.

Appendix A Detailed analytical calculations for generalized Werner states and optimal measurement angles

a.1 Projective measurement outcomes

When applying to the generalized Werner state ρ defined in Eq. (2) a projective measurement described by an observable (i.e. a measurement setting) A⊗B, with

U=cosθUσz+sinθU(cosϕUσx+sinϕUσy),

(6)

where σx, σy and σz are the Pauli matrices and U=A,B, the computational basis states are transformed as:

|0⟩U

→

cosθU2|0⟩U+sinθU2eiϕU|1⟩U,

|1⟩U

→

−sinθU2|0⟩U+cosθU2eiϕU|1⟩U,

and the probabilities of the four possible combinations of outcomes a for A and b for B are:

P(aA,aB)=⟨ab|(A⊗B)ρ(A⊗B)†|ab⟩,

which gives:

P(0A,0B)

=

14+p4[(2α−1)cosθB+(2α−1+cosθB)cosθA

+

(2η−1)√α(1−α)cos(φ−ϕA−ϕB)sinθBsinθA],

P(0A,1B)

=

14+p4[(1−2α)cosθB−(1−2α+cosθB)cosθA

−

(2η−1)√α(1−α)cos(φ−ϕA−ϕB)sinθBsinθA],

P(1A,0B)

=

14+p4[(2α−1)cosθB−(2α−1+cosθB)cosθA

−

(2η−1)√α(1−α)cos(φ−ϕA−ϕB)sinθBsinθA],

P(1A,1B)

=

14+p4[(1−2α)cosθB+(1−2α+cosθB)cosθA

+

(2η−1)√α(1−α)cos(φ−ϕA−ϕB)sinθBsinθA],

The correlation function of this measurement is

E(A⊗B)

=

P(aA=bB)−P(aA=¯bB)

=

P(0A,0B)+P(1A,1B)−P(0A,1B)−P(1A,0B).

a.2 Visibilities Vz and Vx

The visibility is Vz=|E(σz⊗σz)|=p in the σz basis and Vx=|E(σx⊗σx)|=2p(2η−1)√α(1−α) in the σx basis.

a.3 Concurrence C

For an X-shaped density matrix of the form

⎛⎜
⎜
⎜⎝a00w0bz00z∗c0w∗00d⎞⎟
⎟
⎟⎠,

Wootters’ concurrence Wootters (1998) can be simply calculated Yu and Eberly (2007) as C=2max(0,|z|−√ad,|w|−√bc), which for the state of Eq. (2) gives Eq. (3):

C=2max(0,p(2η−1)√α(1−α)−1−p4).

a.4 Bell-CHSH parameter S

The Bell parameter for the CHSH inequality S≤2Clauser et al. (1969) is given by:

S

=

|E(A⊗B)−E(A⊗B′)+E% (A′⊗B)+E(A′⊗B′)|

=

p(cosθA(cosθB−cosθB′)+cosθA′(cosθB+cosθB′)

+2(2η−1)√α(1−α)cosφ

where A,A′,B,B′ are four measurement settings described by Eq. (6) with ϕA=ϕA′=ϕB=ϕB′=0 (although one could use different values of ϕ in case φ≠0,π).
The optimal measurement angles do not depend on the value of p and can be calculated as in Ref. Gisin (1991): fixing θA=0 (A=σz) and θA′=π/2 (A′=σx), the optimal measurement angles for Bob (that maximize S) are:

θB,B′=2atan(√4(2η−1)2α(1−α)+1±12(2η−1)√α(1−α)),

(7)

and the corresponding maximal value of the Bell parameter (Eq. (4)) is:

S=2p√1+4(2η−1)2α(1−α).

a.5 Fine-grained steering parameter F

The fine-grained steering parameter given in Eq. (LABEL:Eq:RhoSteeringPr) of the main text is calculated as follows:

F=maxa,b(Fa,b)=max(F0,0,F1,1),

with Fa,b=P(bP|aS)+P(bQ|aT) and P(bB|aA)=P(aA,bB)/(P(aA,bB)+P(aA,¯bB)), with A=S,T and B=P,Q.
Thus, if we impose that θQ=θP+π/2 and ϕQ=ϕP=0 (one possible choice for two maximally incompatible measurement operators P and Q for Bob), we have, for any choice of θP:

F0,0

=

[1+p[(2α−1)cosθP+(2α−1+cosθP)cosθS

+2(2η−1)√α(1−α)cos(φ−ϕS)sinθPsinθS]]

×14[1+p(2α−1)cosθS]

+

[1+p[−(2α−1)sinθP+(2α−1−sinθP)cosθT

+2(2η−1)√α(1−α)cos(φ−ϕT)cosθPsinθT]]

×14[1+p(2α−1)cosθT],

and

F1,1

=

[1+p[(1−2α)cosθP+(1−2α+cosθP)cosθS

+2(2η−1)√α(1−α)cos(φ−ϕS)sinθPsinθS]]

×14[1+p(1−2α)cosθS]

+

[1+p[−(1−2α)sinθP+(1−2α−sinθP)cosθT

+2(2η−1)√α(1−α)cos(φ−ϕT)cosθPsinθT]]

×14[1+p(1−2α)cosθT].

F0,0 is maximized for

⎧⎪⎨⎪⎩XsinθP[cosθS+p(2α−1)]+YcosθPsinθS=0XcosθP[cosθT+p(2α−1)]−YsinθPsinθT=0,=
which gives, for θP≠0,π:

θS=2

atan

[1XsinθP[1−p(2α−1)]×(YcosθP

+√X2sin2θP[1−p2(2α−1)2]+Y2cos2θP)],

θT=2

atan

[1XcosθP[1−p(2α−1)]×(−YsinθP

+√X2cos2θP[1−p2(2α−1)2]+Y2sin2θP)],

and F1,1 is maximized for

⎧⎪⎨⎪⎩XsinθP[cosθS−p(2α−1)]+YcosθPsinθS=0XcosθP[cosθT−p(2α−1)]−YsinθPsinθT=0,=
which gives, for θP≠0,π:

θS=2

atan

[1XsinθP[1+p(2α−1)]×(YcosθP

+√X2sin2θP[1−p2(2α−1)2]+Y2cos2θP)],

θT=2

atan

[1XcosθP[1+p(2α−1)]×(−YsinθP

+√X2cos2θP[1−p2(2α−1)2]+Y2sin2θP)],

with ϕS=ϕT=φ, X=(2η−1)√α(1−α) and Y=[p(2α−1)2−1]/2.

Figure 6: Value of F as a function of θP and p for 6 different values of α, with η=1. The black line in the green region on each plot corresponds to F=3/4.

We remark that

F=F0,0 for θP∈[0;π4] and α∈[12;1] or for θP∈[π4;π] and α∈[0;12],

F=F1,1 for θP∈[0;π4] and α∈[0;12] or for θP∈[π4;π] and α∈[12;1],

cos(θP+π4)≥0 for θP∈[0;π4] and cos(θP+π4)<0 for θP∈]π4;π],

2α−1=|2α−1| for α∈[12;1] and 2α−1=−|2α−1| for α∈[0;12],

1−2α=|2α−1| for α∈[0;12] and 1−2α=−|2α−1| for α∈[12;1].

Thus, with σ=sign(cos(θP+π4)), we obtain the general expression of the fine-grained steering parameter (Eq. (LABEL:Eq:RhoSteeringPr)):

F

=

[1+p[σ|2α−1|cosθP+(σ|2α−1|+cosθP)cosθS

+2(2η−1)√α(1−α)cos(φ−ϕS)sinθPsinθS]]

×[4[1+pσ|2α−1|cosθS]]−1

×

[1+p[−σ|2α−1|sinθP+(σ|2α−1|−sinθP)cosθT

+2(2η−1)√α(1−α)cos(φ−ϕT)cosθPsinθT]]

×[4[1+pσ|2α−1|cosθT]]−1,

for the optimal angles (for θP≠0,π)

θS

=

2atan[1Xsin(θP)[1−pσ|2α−1|]×(Ycos(θP)

(8)

+√X2sin2(θP)[1−p2(2α−1)2]+Y2cos2(θP))],

θT

=

2atan[1Xcos(θP)[1−pσ|2α−1|]×(−Ysin(θP)

+√X2cos2(θP)[1−p2(2α−1)2]+Y2sin2(θP))],

or (for θP=0,π)

θS

=

θP,

(10)

θT

=

σacos(−pσ|2α−1|).

(11)

All the analytical expressions of optimal angles were calculated with Wolfram AlphaWol .

In Fig. 6, we plot the value of F as a function of the measurement angle θP and the noise parameter p for different values of α, with η=1. For these plots, θS and θT have been optimized as above and θQ=θP+π/2. We see that for a given value of p, the minimum value of F is obtained for θP=π/4, except for α=1/2 for which every choice of θP gives the same value of F for a given value of p. The steering bound on p as a function of α (for Scenario II) given in Fig. 2 and 3 corresponds to the value of p for which F=3/4 with θP=π/4.

c.1 Probabilities

The joint probabilities P(aA,bB) are experimentally estimated from coincidence counting of single-photon detection events in both avalanche photodiodes of the source presented in Fig. 1. For each joint measurement setting A⊗B, four such coincidence counts C(ϑA,ϑB) are recorded with Alice’s and Bob’s polarizers set at the angles {ϑA;ϑB}={θA2;θB2}, {θA2;θB2+π2}, {θA2+π2;θB2} and {θA2+π2;θB2+π2}, corresponding to the measurement outcomes {0A;0B}, {0A;1B}, {1A;0B} and {1A;1B} respectively. Thus the experimental coincidence probabilities of obtaining the outcomes a and b for the measurement setting A⊗B are: